For realα(α>1), we introduce subclassesM(α)andN(α)of analytic functionsf(z)withf(0)=0andf′(0)=1inU. The object of the present paper is to consider the coefficient inequalities for functionsf(z)to be in the classesM(α)andN(α). Further, the bounds ofαfor functionsf(z)to be starlike inUare considered.

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Abstract. Applying the coefficient inequalities for functions f (z) belonging to a certain class M D(α, β) of normalized analytic functions in the open unit disk U , two subclasses M 1 (α, β) and M 2 (α, β) are defined. The object of the present paper is to derive some properties for functions f (z) in the classes M 1 (α, β) and M 2 (α, β) involving their generalized convolution by utilizing methods based upon the Hölder-type inequalities.Mathematics subject classification (2000): 30C45, 30A10, 26D07.

Let𝒮*(p,α)be the class of functionsf(z)which are analytic andp-valently starlike of orderαin the open unit diskE. The object of the present paper is to derive an interesting condition forf(z)to be in the class𝒮*(p,α).

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